JP4775596B2 - Calculation of filter coefficient used for equalizer of communication receiver - Google Patents

Description

  The present invention generally relates to equalizers in wireless communication receivers.

  Most modern wireless communication systems transmit data over a dispersive communication channel that varies over time. Among the distortions introduced by the channel, inter-symbol interference (ISI) is important because it severely degrades the performance of the receiver. To mitigate the effects of ISI, many receivers use equalizers. The general architecture of an equalizer has a filter, an adder for synthesizing the output of the filter, and a decision device. The filter is a linear finite-impulse-response (FIR) having complex coefficients. The determination device performs an operation on the complex number input and outputs a complex value representing a signal constellation point of the modulation scheme.

  In general, the equalizer filter coefficients are optimized together according to criteria appropriate for the communication system. Determining the optimal equalizer filter coefficients is a computationally intensive task because it requires a solution to a large set of linear equations. Today, two common methods are commonly used. The first method is an adaptive approach, and the second method is a direct matrix inversion approach.

In the adaptive method, the equalizer filter coefficients are first set to some initial values. The output error signal, defined as the difference between the input and output of the equalizer decision device, is then used to recursively adjust the equalizer filter coefficients to an optimal setting. Depending on the coefficient adaptation algorithm used, a training sequence may be required. A training sequence is a known set of codes that a transmitter transmits with data. In US Pat. No. 5,068,873 to Murakami, a least mean square (LMS) or Kalman filter algorithm is used for adaptation. This method requires a training sequence. The LMS algorithm requires O (N) complex operations for each iteration. Here, N is the total number of coefficients to be optimized. In addition, a large number of iterations (>> N) are required to converge the equalizer filter coefficients to the optimum value. The Kalman filter algorithm converges faster with an optimal solution, but requires O (N ² ) operations per iteration. Similarly, US Pat. No. 5,283,811, granted to Chennankeshu et al., Uses a fast Kalman algorithm for coefficient adaptation of a decision feedback equalizer (DFE). Yes. US Pat. No. 3,974,449 to Falconer describes a DFE adaptation method that does not use a training sequence.

In the direct matrix inversion method, the channel response to the signaling pulse is first estimated. This estimate is the response of the channel filtered by the receiver filter to the transmitter spectral shaping pulse. Next, an equalizer coefficient is obtained from an estimate of the response of the channel to the signaling pulse by solving a set of complex-valued linear equations. In general, solving these equations requires inversion of an N × N square matrix, which requires O (N ³ ) complex multiplications. U.S. Pat. No. 5,436,929 to Kawas Kaleh uses the positive definite nature of square matrices and Hermitian symmetry so that Cholesky decomposition can be used. . The Cholesky factorization requires O (N ³ ) complex multiplications to factor a positive Hermitian symmetric matrix into the product of a lower triangular matrix and an upper triangular matrix. The upper triangular matrix is equivalent to the Hermitian transpose of the lower triangular matrix. The triangular matrix can be easily inverted and requires O (N ³ ) multiplications. US Pat. No. 5,790,598 issued to Moreland et al. Describes a recursive method using Cholesky decomposition. Both of these techniques still require O (N ³ ) complex multiplications.

In general, using the direct matrix inversion method, the optimization of equalizer coefficients requires at least O (N ³ ) complex multiplications. This complexity makes it impractical to implement this method in many real communication systems. If a large number of iterations are required, this complexity can be even greater for adaptive methods. In addition, the adaptive method usually yields a suboptimal optimal solution compared to the direct matrix inversion method.

  In addition, the mathematical notation used in this specification is an uppercase notation, for example, W is used to indicate a matrix, and a lowercase notation, for example, w is used to indicate a vector.

FIG. 1 shows a typical equalizer device 10 used in a communication system. The equalizer 10 is designed to recover a transmitted signal distorted by a dispersive channel (characterized by the channel response matrix H) and noise (characterized by the variance σ ² ). In order to calculate the filter coefficients for the equalizer, a channel response matrix G is calculated based on an estimate of the channel response matrix H. This step is generally performed using pilot signaling. Signal processing is normally performed inside the equalizer 10 for each block. The first channel matrix estimation block 12 receives an input signal (vector r) for use in estimating the channel response matrices H and G. Next, in a filter coefficient calculation block 14, a filter coefficient vector is calculated, where the calculation includes an inversion of the channel response matrix G. The calculated filter vector w ₀ is then used by the FIR filter 16 to equalize the received vector r to obtain output data that is subsequently used in the communication receiver.

  In order to understand the computational complexity of the inversion of the channel response matrix G, an exemplary method for matrix inversion will now be described.

  The channel estimate vector h is used for the calculation of the vector f. Here, for example, for five multipaths and three timing positions for each multipath,

And The matrix G is constructed as follows.

Since the channel response matrix G is Hermitian and positive, there is a unique lower (upper) triangular matrix L (U) such that G = LL ^H = U ^H U, where the superscript H indicates Hermitian transpose of the matrix. In general, we obtain

GG ⁻¹ = I⇔L (L ^H G ⁻¹ ) = I⇔LD = I (1)
Here, the superscript -1 indicates the inverse of the matrix, and I is the unit matrix. Furthermore, the following formula is obtained.

L ^H G ^-1 = D (2)
To find the channel response inverse matrix G ⁻¹ , the following steps need to be performed.

Step 1: Cholesky decomposition of the channel response matrix is performed to obtain a lower triangular matrix L. - computational complexity O (N ^3).

  Step 2: Solve equation (1) using forward substitution as lower triangular matrix L to obtain matrix D.

Step 3: Solve Eq. (2) using reverse permutation for matrices D and L ^H to obtain the inverse channel response matrix G ⁻¹ .

Since the simultaneous linear equations have N times N unknowns (D and G ⁻¹ are N × N matrices), the computational complexity of steps 2 and 3 of the above method is O (N ³ ). .

As described in co-pending Australian Patent Application No. 2005203278 entitled "Calculation Method for Filter Coefficients of Communication Receiver Equalizer" filed July 26, 2005 under the name of NEC Australia Pty. Using the method, the computational complexity of steps 2 and 3 can be reduced to O (N ² ). In this case, the complexity of the decomposition step (step 1) is a major contributor to complexity, processing time and size in the calculation of the equalizer filter coefficients in the equalizer.
US Pat. No. 5,068,873 US Pat. No. 5,283,811 U.S. Pat. No. 3,974,449 US Pat. No. 5,436,929 US Pat. No. 5,790,598 WO2006 / 016722

  Therefore, there is a need for an efficient method for calculating the equalizer filter coefficients in the equalizer that can be practically implemented inside the communication receiver. A method for calculating equalizer filter coefficients that is less complicated to compute than currently known methods is desirable. It is further desirable to provide a method for calculating equalizer filter coefficients with fewer calculations that can lead to reduced circuit area and power consumption and / or faster processing times. It is further desirable to provide a method for calculating equalizer filter coefficients for an equalizer that improves or overcomes one or more of the problems of known coefficient calculation methods.

  In a first aspect, the present invention provides a method for calculating equalizer filter coefficients for a communication receiver based on a real matrix T generated from a channel estimation vector f derived from a channel estimation input. This avoids the need to invert the complex channel matrix G.

  The method preferably includes forming T directly from the vector f.

In an embodiment where N is chosen to be odd and f is a vector of dimension N as defined above and T is an N × N matrix, the middle column of the inverse matrix of G ( The matrix T can be generated by solving Gc ₀ = i ₀ for middle column) c ₀ and each of the real and imaginary terms in the middle column i ₀ of the unit matrix. In this embodiment, T can be generated as follows:
T ((N + 1) / 2, (N + 1) / 2) = R [f (1) / 2]. Here, the rows and columns are given numbers from 1 to N, the notation T (x, y) indicates the entry of the x th row and the y th column of the matrix T, and R [z] is the real number of z And I [z] represents the imaginary part of z;
For row (N + 1) / 2, column number 1 to (N-1) / 2,
T ((N + 1) / 2, (N + 1) / 2-column number) = R [f (column + 1)]. And,
T ((N + 1) / 2, (N + 1) / 2 + column number) = I [f (column + 1)],
From row number 1 to (N-1) / 2 and for the middle column,
T (line number, (N + 1) / 2) = R [f ((N + 1) / 2)], and
T (N + 1−line number, (N + 1) / 2) = I [f ((N + 1) / 2)],
For row numbers 1 to (N-1) / 2 and column numbers 1 to (N-1) / 2,
T (row number, column number) = R [f (column number)] + R [f (N + 1−column number)], and
T (N + 1−row number, N + 1−column number) = R [f (column number)] − R [f (N + 1−column number)],
For row numbers 1 to (N−1) / 2 and column numbers ((N + 1) / 2 + 1) to N,
T (row number, column number) = I [f (column number)] − I [f (N + 1−column number)], and
T (N + 1−row number, N + 1−column number) = I [f (column number)] + I [f (N + 1−column)],
For column numbers N to 2 of the f vector,
f (col) = f (column-1),
R [f (1)] = T ((N + 1) / 2, (N + 1) / 2−row),
I [f (1)] = − T ((N + 1) / 2, (N + 1) / 2 + row)).

This method
Performing matrix inversion on T to determine the middle column of T ⁻¹ ;
Forming a vector v from the middle column of T ⁻¹ and a constant;
determining c ₀ ^H based on v;
Determining the filter coefficient w ₀ using the relationship w ₀ = c ₀ ^H H ^H ;
Where H ^H is the Hermitian transpose of the channel response matrix H.

  In an exemplary embodiment, matrix inversion involves performing a Cholesky decomposition on T to obtain a lower triangular matrix L, and then performing forward and reverse permutations on L; Can be included.

If v and c ₀ ^H have elements from 1 to N and the notation v (x) indicates the xth element of the vector v, then determining c ₀ ^H based on v is
setting c ₀ ^H ((N + 1) / 2) equal to v ((N + 1) / 2);
for elements from c ₀ ^H 1 to (N-1) / 2, c 0 H ( element number) and v (element number) - (0 + i (v (N + 1- element number)) set equal to the When,
for elements from c ₀ ^H of (N + 1) / 2 to N, and setting equal c ₀ ^H a (element number) to v (N + 1-element number) + (0 + i (v (element number)),
It is desirable to include.

In a second aspect, the present invention provides a method for processing a received signal at a communication receiver, the method comprising:
Calculating one or more equalizer filter coefficients using the method of the first aspect of the invention;
Equalizing the received signal using the calculated equalizer filter coefficients;
Is included.

  In a third aspect, the present invention provides an equalizer configured to implement a method according to any of the above aspects according to the present invention.

In a fourth aspect, the present invention provides an equalizer for use in a communication receiver, the equalizer comprising:
A first input for receiving input data for equalization;
A second input for receiving channel estimation data for the input data;
A signal processing path configured to perform the signal processing method according to claim 8 on input data and generate equalized output data;
At least output for outputting equalized output data;
Is included.

The equalizer is:
A channel matrix calculation block configured to calculate vectors h and f based on a basis of received channel estimates;
A real matrix computation block configured to form a matrix T;
A matrix decomposition block for performing matrix decomposition on T;
Forward and backward permutation blocks configured to calculate the values of the elements in the middle column of T ^-1 ;
A central column calculation block configured to determine element values in the central column (c ₀ ) of G ⁻¹ ;
A filter coefficient generation block configured to calculate a filter coefficient w ₀ = c ₀ ^H H ^H , and a finite impulse response filter;
One or more of each signal processing block.

  In yet another aspect, the present invention provides a communication receiver including an equalizer according to the previous aspect of the invention.

  To facilitate understanding of the present invention, reference is made herein to the accompanying drawings, which illustrate an equalizer and a method for calculating equalizer filter coefficients in the preferred embodiment. It should be understood that the present invention is not limited to the preferred embodiments illustrated in the accompanying drawings.

The computation of the vector of filter coefficients is aimed at finding the middle row w ₀ of the matrix W = [H ^H HH + σ ² I] ⁻¹ H ^H = G ⁻¹ H ^H. Here, G and H are channel response matrices, I is a unit matrix, the superscript H indicates the Hermitian transpose of the matrix, and the superscript -1 indicates the inverse or inverse of the matrix.

This means that in order to obtain the vector w ₀ , w ₀ = r ₀ H ^H , so the central row vector r ₀ of G ⁻¹ must be calculated and then multiplied by H ^H. To do. If the dimension of G is N, it will require O (N ³ ) order complex multiplications to calculate r ₀ .

The present invention utilizes the fact that G ⁻¹ and G are Hermitian matrices. Thus, instead of having to calculate G ⁻¹ completely and then find its central row r ₀ , only the central column of G ⁻¹ (denoted as c ₀ ) is calculated, It is used to obtain _{^{w 0 = c 0 H H H}} .

Furthermore, as a result of G ^-1 being Hermitian, the inventor also notices that G ^-1 is also Toeplitz and that the central element of c ₀ is a real number, so that c ₀ is , (N + 1) / 2 real terms and (N-1) / 2 imaginary terms. These N real variables constitute the vector v.

If the central column i ₀ of the unit matrix I is considered as a complex number from the definition GG ⁻¹ = I, the following equation is obtained.

Gc ₀ = i ₀
Solving the above matrix equations for each of the real and imaginary terms in i ₀ yields N different real equations including N real variables of v. This can be written as Tv = i.

Where T is a new N × N real matrix made from the coefficients of v in the N different real equations above, and i is made from the appropriate real and imaginary terms in i ₀ . Is a vector. Therefore, v = T ⁻¹ i.

Since all elements of i are zero except the central element, v is a constant multiple of the central column of T- ¹ . Once the real vector v is known, c ₀ can be known. This result changes the original problem of performing inversion of the G matrix with complex numbers to real number inversion of the T matrix performed as shown below.

To help understand the preferred embodiment, an example of how the matrix T can be formed will now be described. In this example, assume N = 3. C ₀ , the inverse central column of matrix G of channel r, is a column vector

Can be expressed as This column vector has three real terms: x1, x2, y1.

  in this case,

far.

Extending Gc ₀ = i _{(N + l) / 2} , for row 1, we get

a * (x1 + jy1) + (b + jc) * x2 + (d + je) * (x1-ji1) = (0 + j0)
If the real and imaginary terms are separated, the following equation is obtained.

Real number: (a + d) * x1 + b * x2 + e * y1 = 0
Imaginary number: e * xl + c * x2 + (ad) * y1 = 0
For row 2, we get

(b−jc) * (x1 + ji1) + a * x2 + (b + jc) * (x1−gy1) (1 + j0)
The real term in this equation is:

2 * b * x1 + a * x2 + 2 * c * y1 = 1
If both sides of this equation are divided by 2, the following equation is obtained.

Real number: b * x1 + (a / 2) * x2 + c * y1 = 0.5
These three equations can then be used to create a basis for simultaneous linear equations as follows:

  In a preferred embodiment, an equalizer device is proposed as shown in the functional diagram in FIG. This type of equalizer device can greatly reduce the computational requirements for generating the filter coefficients (on the order of 4) and is therefore suitable for use in an actual communication system.

  The equalizer 200 has seven main blocks as follows.

  Channel matrix calculation block 202: This block calculates the vectors h and f based on the channel estimation input as described above.

  Real Matrix Calculation Block 204: This block uses the vector f to directly form the matrix T instead of the G matrix described above.

  An exemplary matlab algorithm that can be used to generate T from the vector f according to the methodology described above is shown in FIG.

  Matrix decomposition block 206: This functional block performs standard matrix decomposition (eg, Cholesky decomposition) on T.

Forward and reverse permutation block 208: This block performs standard forward and reverse permutations to obtain the middle column of T- ¹ .

G ^-1 (c ₀ ) middle column calculation block 210: This block forms v as a constant (0.5) times the middle column of T ^-1 . Knowing the real vector v will know c ₀ and therefore c ₀ ^H. The exemplary matlab algorithm of FIG. 4 can be used to calculate c ₀ and thus c ₀ ^H.

Filter coefficient generation block 212: This block performs calculations to obtain the filter coefficient w ₀ = c ₀ ^H H ^H.

FIR filter 214: The FIR filter uses the updated filter coefficient w ₀ to filter the input data from time to time.

  As will be appreciated by those skilled in the art, the illustrated embodiment operates on a real matrix T instead of a complex matrix G, so the computational complexity, size and processing time are greatly reduced, As a result, the equalizer made according to this embodiment is implemented in a practical communication device.

  The invention disclosed and defined herein may be referred to in this specification or the accompanying drawings, or extended to all other combinations of two or more individual features that are apparent from this specification or the accompanying drawings. Will be understood. All these different combinations constitute various alternative aspects of the invention.

  As used herein, the term “having” (or grammatical variations thereof) is equivalent to the term “including” and should not be interpreted as excluding the presence of other elements or features; It will be further understood.

1 is a schematic diagram of a known equalizer used in a communication system. FIG. 3 is a schematic diagram of an equalizer for calculating equalizer filter coefficients according to an embodiment of the present invention. Figure 7 is a list of exemplary matlab implementations of algorithms that can be used to generate T from a vector f according to embodiments according to the present invention. FIG. 6 is a list of exemplary matlab implementations of algorithms that can be used to calculate c ₀ in an embodiment according to the present invention.

Claims (7)

A method of calculating equalizer filter coefficients in a communication receiver,
Determining a channel estimation vector f from a vector h of at least one channel estimation value based on a signal received at the communication receiver ;
Determining equalizer filter coefficients based on a real matrix T generated from the channel estimation vector f;
Only including,
Assuming that f is a vector of dimension N and n ≧ 0 and the nth element of the vector h is represented by h [n], the element f (i) of the vector f is

Represented by
When N is an odd number and T is an N × N matrix, the method
c further includes generating a matrix T by solving the Gc ₀ = i ₀ for each real term and the imaginary term in the _0, where c ₀ is the inverse of the center column of the channel response matrix G I ₀ is the middle column of the identity matrix,
The rows and columns of the matrix T and the columns of the vector f are numbered from 1 to N, the notation T (x, y) indicates the entry of the x th row and y th column of the matrix T, and the notation f ( y) denotes the entry of the y th column in the vector f, R [z] denotes the real part of the number z, and I [z] denotes the imaginary part of the number z,
The elements of the matrix T are
T ((N + 1) / 2, (N + 1) / 2) = R [f (1) / 2],
For row (N + 1) / 2, column number 1 to (N-1) / 2,
T ((N + 1) / 2, (N + 1) / 2−column number) = R [f (column + 1)], and
T ((N + 1) / 2, (N + 1) / 2 + column number) = I [f (column + 1)],
From row number 1 to (N-1) / 2 and for the central column,
T (line number, (N + 1) / 2) = R [f ((N + 1) / 2)], and
T (N + 1−line number, (N + 1) / 2) = I [f ((N + 1) / 2)],
For row numbers 1 to (N-1) / 2 and column numbers 1 to (N-1) / 2,
T (row number, column number) = R [f (column number)] + R [f (N + 1−column number)], and
T (N + 1−row number, N + 1−column number) = R [f (column number)] − R [f (N + 1−column number)],
For row numbers 1 to (N−1) / 2 and column numbers ((N + 1) / 2 + 1) to N,
T (row number, column number) = I [f (column number)] − I [f (N + 1−column number)], and
T (N + 1−row number, N + 1−column number) = I [f (column number)] + I [f (N + 1−column)],
For column numbers N to 2 of the f vector,
f (column) = f (column-1),
R [f (1)] = T ((N + 1) / 2, (N + 1) / 2−row),
I [f (1)] = − T ((N + 1) / 2, (N + 1) / 2 + row))
Generated as a method. The method further comprises:
Performing matrix inversion on T to determine the middle column of T ⁻¹ ;
Forming a vector v from the middle column of T ⁻¹ and a constant;
determining c ₀ ^H based on v;
Determining the filter coefficient w ₀ using the relationship w ₀ = c ₀ ^H H ^H ;
Only including, where H ^H is Ri Hermitian transpose der estimated channel response matrix H based on signals received by the communication receiver,
v and c ₀ ^H have elements from 1 to N, notation v (x) and c ₀ ^H (x) as respectively the x th element of the vector v and c ₀ ^H, of c ₀ ^H The element is based on v
c ₀ ^H ((N + 1) / 2) is equal to v ((N + 1) / 2),
for elements from 1 c ₀ ^H to (N-1) / 2, c 0 H (x) is v (x) - (equal to 0 + i (v (N + 1-x)), and
The c ₀ ^H (x) is determined to be equal to v (N + 1−x) + (0 + i (v (x)) for the elements from (N + 1) / 2 to N of c ₀ ^H. The method according to 1 . A method of processing a received signal in a communication receiver,
Using the method of claim 1 or 2 to calculate one or more equalizer filter coefficients;
Equalizing the received signal using the calculated equalizer filter coefficients;
Including methods. An equalizer for use in a communication receiver, wherein the equalizer is configured to implement a method comprising the method of any one of claims 1 to 3 . An equalizer used for a communication receiver,
A first input for receiving input data for equalization;
A second input for receiving channel estimation data for the input data;
A signal processing path configured to perform the signal processing method according to claim 3 on the input data and generate equalized output data;
At least an output for outputting the equalized output data;
Including equalizer. The equalizer according to claim 4 or 5, wherein :
A channel matrix calculation block configured to calculate vectors h and f based on a basis of received channel estimates;
A real matrix computation block configured to form a matrix T;
A matrix decomposition block for performing matrix decomposition on T;
Forward and backward permutation blocks configured to calculate the values of the elements in the middle column of T ^-1 ;
A central column calculation block configured to determine element values in the central column (c ₀ ) of G ⁻¹ ;
A filter coefficient generation block configured to calculate a filter coefficient w ₀ = c ₀ ^H H ^H , and a finite impulse response filter;
One or more containing, etc encoder of the signal processing blocks. Communications receiver including an equalizer according to item 1 to any one of claims 4 to 6.

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